Factorising Quadratics into Two Brackets
Undo the expansion of two brackets for quadratics of the form x² + bx + c by finding the number pair that multiplies to c and adds to b. Check every answer by expanding it back.
What a learner can do afterwards
- Factorise x² + 7x + 12 by finding the pair that multiplies to 12 and adds to 7
- Choose signs correctly for a quadratic with a negative constant, such as x² - 2x - 15
- Check a factorisation by expanding the brackets back to the original quadratic
1 · Read
Expanding double brackets turned (x + 3)(x + 5) into x squared + 8x + 15. Factorising runs that same move in reverse. You are handed the expanded answer and have to find the two brackets that made it. The skill pays off right away, because factorised form is the key that opens quadratic equations, which are next on the thread.
Here is the rule that does the work. The pair always multiplies to c and adds to b. The signs fall out of that. If c is positive, both numbers in the pair carry the sign of b. If c is negative, the pair has different signs, and the bigger number takes the sign of b.
Factorise x squared + 8x + 15. Here b is 8 and c is 15, so the pair is two positive numbers. The factor pairs of 15 are 3 and 5, or 1 and 15. Only 3 and 5 adds to 8, so the brackets are (x + 3)(x + 5). Now the check: expand back. x times x gives x squared, the outer and inner terms give 5x and 3x, and 3 times 5 gives 15. The x terms collect to 8x, so we land exactly on the original.
Now a negative constant. Factorise x squared + 2x - 15. Here b is 2 and c is -15, so the pair has different signs. The candidates are 5 and -3, or 3 and -5. Only 5 and -3 adds to 2, so the brackets are (x + 5)(x - 3). Check by expanding: the outer and inner terms give 5x and -3x, which collect to 2x, and 5 times -3 gives -15. We land back on the original.
The order of the brackets never matters. (x + 3)(x + 5) and (x + 5)(x + 3) are the same answer, because multiplying has no order. Write whichever pair feels natural.
Two habits to keep. If no pair of integers does both jobs, the quadratic cannot be factorised over the integers, and saying so is the correct answer. And make the check automatic: expand the brackets back and confirm you land on the original every time.
Find the pair that multiplies to c and adds to b, put it in brackets with x, then check by expanding back out.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.